We can evalute this series
(1/2) + (1/2)^2 + (1/2)^3 + (1/2)^4 + (1/5)^5 + (1/2)^6 + (1/2)^7 + (1/2)*8 + .......
The sum of this geometric series = (a) / ( 1 - r) where a is the first term and r is the common ratio
The sum is (1/2) / (1 - 1/2) = (1/2) / (1/2) = 1
And from this sum we can subtract this sum
(1/2)^3 + ( 1/2)^6 + (1/2)^9 + (1/12)^12 + ..........
The first term = (1/2)^3 and the common ratio = (1/2)^3
( 1/2)^3 / [ 1 - (1/2)^3] = (1/8) / [ 1 - 1/8] = (1/8) / (7/8) = 1/7
So....the given series evaluates to
1 - 1/7 =
6/7
